Real Analysis/Introduction

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Real Analysis Introduction

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Introduction

The subject of real analysis is concerned with studying the behavior and properties of functions, sequences, and sets on the Real number line, which we denote $\mathbb {R}$. Concepts that we wish to examine through real analysis include properties like Limits, Continuity, Derivatives (rates of change), Integration (amount of change over time). Many of these ideas that, on a conceptual or practical level, are dealt with at lower levels of mathematics, including a regular First-Year Calculus course, and so to the uninitiated reader the subject of Real Analysis may seem rather shallow and trivial. However, the depth and complexity (and arguably the beauty) of Real Analysis is that we wish to generalize these properties away from the "nice" functions and sets dealt with in everyday mathematics, and rigorously prove these properties for all objects in the universe of the Real numbers. Thus real analysis can to some degree be viewed as a development of a rigorous, well-proven framework to support spatial and conceptual ideas that we frequently take for granted.

Real Analysis is a very straightforward subject, in that it is simply a nearly linear development of the ideas mentioned above. However, as the object of Real Analysis is to make things that we may already "know" more rigorous and definite, we cannot begin our development on unproven assumptions. Thus the approach we take to set this matter straight is to define the real numbers axiomatically. In layman's terms, we set down the properties which we think define the real numbers. We then prove from these properties and these properties only that the real numbers behave in the way which we have come to understand that regular objects in space behave. Lastly, we build a number system, and show it satisfies these properties.

From that foundation, we will develop first a set of results about the real number line, and then move on to results on a two-dimensional plane of real numbers, and then eventually we will generalize many of our results into n-dimensions.

Note: A table of the math symbols used below and their definitions is available here.